Bounding the joint numerical range of Pauli strings by graph parameters

Abstract

The interplay between the quantum state space and a specific set of measurements can be effectively captured by examining the set of jointly attainable expectation values. This set is commonly referred to as the (convex) joint numerical range. In this work, we explore geometric properties of this construct for measurements represented by tensor products of Pauli observables, also known as Pauli strings. The structure of pairwise commutation and anticommutation relations among a set of Pauli strings determines a graph $G$, sometimes also called the frustration graph. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range. Such an outer approximation can be very practical since ellipsoids can be handled analytically even in high dimensions. We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O. Gühne, arXiv:2207.02197], and answer an open question in [M. B. Hastings and R. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graph parameter that we call $\beta (G)$. Besides, we develop this approach in different directions, such as comparison with graph-theoretic approaches in other fields, applications in quantum information theory, numerical methods, properties of the new graph parameter, etc. Our approach suggests many open questions that we discuss briefly at the end.

14+3 pages, 5 figures

References in corpus (5)

• The computational difficulty of finding MPS ground states
• Higher entropic uncertainty relations for anti-commuting observables
• Product numerical range in a space with tensor product structure
• A Geometric Algebra Perspective On Quantum Computational Gates And Universality In Quantum Computing
• Numerical shadows: measures and densities on the numerical range

Cited by in corpus (6)

• Uncertainty relations from state polynomial optimization
• Bounds on the ground state energy of quantum $p$-spin Hamiltonians
• An Effective Way to Determine the Separability of Quantum State
• Optimizing Sparse SYK
• Frustration graph formalism for qudit observables
• Positivity of state, trace, and moment polynomials, and applications in quantum information